Combined variation describes the relationship between three or more
variables that vary directly and inversely with one another.
LEARNING OBJECTIVE [ edit ]
Apply the techniques learned with direct and inverse variation to combined variation
KEY POINTS [ edit ]
There must be a minimum of three related variables for their relationship to be one of combined
variation.
Among the three or more related variables, one must directly vary with another and inversely
vary with a third in order for the relationship to be one of combined variation.
An example of combined variation in the physical world is the Combined Gas Law, which relates
pressure, temperature, volume, and moles (amount of molecules) of a gas.
TERMS [ edit ]
constant
An identifier that is bound to an invariant value.
directly proportional
If one variable is always the product of the other and aconstant, the two are said to be directly
proportional.
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Combined variation is used to describe the relationship between three or more variables that
vary directly and inversely with one another. Before go deeper into the concept of combined
variation, it is important to first understand what direct and inverse variation mean.
Direct and Inverse Variation
Simply put, two variables are in direct
variation when the same thing that
happens to one variable happens to the
other. If x and y are in direct variation,
and x is doubled, then y would also be
doubled. The two variables may be
considered directly proportional.
Two variables are said to be in inverse
variation, or are inversely proportional,
when an operation of change is performed
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on one variable and the opposite happens
to the other. For example, if x and y are inversely proportional, if x is doubled, then y is
halved. Combined Variation
To have variables that are in combined variation, theequation must have variables that are in
both direct and inverse variation, as shown in the example below.
Consider the equation:
z = k( xy )
where x, y, and z are variables and k is a constant known as the proportionality constant.
In this example, z varies directly as x and inversely as y.
Given values for any three of x, y, z, and k, the fourth can be found by substitution. For
example, if z=12, x=4 and y=2, we can solve for k:
12 = k
4
2
k=6
Practical Application
A practical example of combined variation is the Combined Gas Law, which relates the
pressure (p), volume (v), moles (n), and temperature (T) of a sample of gas:
P V = nRT
where R is a constant .
Temperature T
Temperature 3T
Illustration of Gay­Lussac's Law, derived from the Combined Gas Law
A constant amount of gas will exert pressure that varies directly with temperature. In this illustration,
volume is held constant by an increased mass weighing down the lid of the container. If not for that extra
mass, the lid would raise, increasing the volume and relieving the pressure.
Solving for P, we can determine the variation of the variables.
P=
nRT
V
In the above equation, P varies directly with n and T, and inversely with V. Thus, pressure
increases as temperature and moles increase. What's more, pressure decreases as volume
increases.